Journal of Public Economics 70 (1998) 5–35
Class systems and the enforcement of social norms q
Harold L. Cole a , George J. Mailath b , *, Andrew Postlewaite b
a
b
Federal Reserve Bank of Minneapolis, Minneapolis, MN, USA
Department of Economics, 3718 Locust Walk, University of Pennsylvania, Philadelphia,
PA 19104 -6297, USA
Abstract
We analyze a model in which there is socially inefficient competition among people. In
this model, self-enforcing social norms can potentially control the inefficient competition.
However, the inefficient behavior often cannot be suppressed in equilibrium among those
with the lowest income due to the ineffectiveness of sanctions against those in the society
with the least to lose. We demonstrate that, in such cases, it may be possible for society to
be divided into distinct classes, with inefficient behavior suppressed in the upper classes but
not in the lower.  1998 Elsevier Science S.A. All rights reserved.
Keywords: Class; Efficiency; Growth; Social competition; Social norms
JEL classification: A10; B40; C73; D31; D90
‘‘Freedom’s just another word for nothin’ left to lose’’ ME AND BOBBY
McGEE (Kris Kristofferson)
‘‘You only have power over people so long as you don’t take everything
away from them. But when you’ve robbed a man of everything, he’s no
longer in your power – he’s free again.’’ (Alexander Solzhenitsyn)
q
Original title: ‘Enforceability of social norms’. The views expressed in this paper are those of the
authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve
System.
*Corresponding author. Fax: 11-215-573-2057; e-mail: gmailath@econ.sas.upenn.edu
0047-2727 / 98 / $ – see front matter  1998 Elsevier Science S.A. All rights reserved.
PII: S0047-2727( 98 )00058-9
6
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
1. Introduction
Conflicts of interest are a ubiquitous aspect of human interactions. A wide
variety of social institutions has evolved to channel and contain these conflicts.
Moreover, this variety potentially explains differences among the economic, as
well as the social, behavior in different societies. It is well understood that
repeated interactions can enhance efficiency by allowing equilibrium behavior that
cannot be optimal in one-off transactions. By and large, the discussions of the
efficiency improvements that are made possible by repeated interactions abstract
from heterogeneities among agents. In this paper, we focus on heterogenous
societies and explain how class systems can naturally ameliorate economic
conflicts in them.
We are interested in social norms of behavior that control, to some extent at
least, economic conflicts. The conflicts we focus on are those in which agents
behave self-interestedly, ignoring the consequences of their behavior on others.
Any social norm that ameliorates such self-interested behavior requires, implicitly
or explicitly, sanctions to be imposed on individuals deviating from the social
norm. However, some individuals may find the benefit in deviating from the
prescribed behavior greater than the cost of the sanctions resulting from a
deviation. While raising the costs is one obvious solution, there are limits. Even
death may not be enough to deter a starving person from stealing bread. Although
not necessary, it will help to focus our discussion to restrict attention to voluntary
associations in which ostracism is the most severe punishment.
A social norm of behavior, then, is limited in its ability to control an
individual’s behavior by what that person has to lose (the size of the sanction) if
he or she deviates from the social norm’s prescriptions. Since ostracism is simply
the exclusion of an individual from the community, the benefits of community
membership and the costs of ostracism are one and the same. Thus, if an
individual gains little or nothing from community membership (in our model, the
agents with the lowest status), the threat of ostracism is of little importance;
consequently, a social norm consistent with optimizing behavior must allow
behavior that is essentially self-interested for such an individual. However, an
individual who benefits substantially from community membership – a high status
individual – can more easily be induced to engage in behavior that is not
self-interested. Consequently, different classes of society may exhibit different
modes of conduct because the prevailing social norm distinguishes among them on
the basis of status.
Our goal is to understand how class structures that arise within a social norm
can affect behavior. If class is to play a meaningful role in understanding behavior,
it cannot be simply as a categorization of people into groups. To define the ‘class
of low income people’ as, say, those people with incomes below half of the
median income has no effect on how those people behave. A change that assigned
more or fewer people to the named ‘class’ would have no effect on how any agent
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
7
behaved or on what he or she received in equilibrium. To be a useful concept,
classes should be more than instruments to describe equilibria. The model we
present will allow us to ask – and answer – questions such as ‘‘What is the
difference between equilibria when there are classes in a society and when there
are no classes?’’ and ‘‘How different would the consumption of an agent be if he
or she were moved from one class to another?’’
The manner in which classes can ameliorate inefficiencies stems from the
existence of multiple equilibria. Standard descriptions of economies are underspecified: a full description of the economic data, including preferences, endowments, and technological possibilities, is not sufficient to uniquely determine the
outcome that arises. Associated with the equilibrium that does arise are social
institutions not normally considered a part of the economic data of the problem
(for example, norms or customs, political and social institutions, etc.). We analyze
a model in which there are multiple equilibria and a subset of the equilibria are
naturally thought of as having a class structure. The equilibria in this subset
qualitatively differ from equilibria with a single class. For example, in an
equilibrium with two classes, there are no interactions between members of one
class and another, yet the behavior of individuals in the upper class qualitatively
differs from what it would be in the absence of the lower class. The mere existence
of the lower class changes the kinds of behavior of individuals in the upper class
that can be supported in an equilibrium. Further, in answer to the second question
in the previous paragraph, in a two class equilibrium, an individual who is moved
from the upper class to the lower will have his or her utility decreased.
We emphasize that our model integrates the notion of class into a traditional
economic model. Individuals do not receive utility from being in one class or
another; they care directly only about the goods they consume. Any concern about
class membership is instrumental: people care about class only to the extent that,
in equilibrium, class affects an individual’s opportunities.
Our starting point is the model of social norms we introduced in Cole et al.
(1992), hereafter CMP. We analyzed in that paper a simple multi-generation model
in which parents could increase their savings in order to improve their children’s
matching prospects. When all families do this, their efforts offset each other,
leaving all parents worse off than if they had not engaged in this bequest
competition. It is not, however, an equilibrium for all families to simply forgo the
effort, because if no other families increase their savings, any single family
benefits by doing so. This competitive over-saving is the self-interested behavior
in the context of CMP. The social competition is essentially a prisoners’ dilemma.
We showed in that paper that in addition to the equilibrium in which this
competitive over-saving behavior occurs, there may exist an equilibrium in which
this socially suboptimal behavior of over-saving was suppressed. This second
equilibrium was supported by an aristocratic social norm that prescribed matching
in a way that was unaffected by savings and imposed sanctions on any family
whose son married better than prescribed by the social norm.
8
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
The equilibrium in which this social competition is not suppressed is similar to
the repeated ‘not cooperate’ equilibrium in the standard repeated prisoners’
dilemma game.1 In the repeated ‘not cooperate’ equilibrium, agents play the
dominating action in each period and so maximize stage game payoffs, yielding a
Pareto inefficient outcome. Just as this equilibrium of the prisoners’ dilemma game
is robust to the number of repetitions, the equilibrium in our model in which social
competition is not suppressed is robust. It is the limit of the unique equilibria of
the corresponding finite horizon models. Other equilibria, in contrast, require some
degree of inter-generational enforcement, since at times a woman must be given an
incentive to reject an offer from a suitor who is wealthier than the woman’s
prescribed match. Such behavior can only be optimal if deviations from the social
norm’s prescription result in future negative consequences (e.g., a loss of status
resulting in the sons of deviating parents being unable to match with any but the
poorest women, as prescribed in the aristocratic social norm). Hence such
equilibria can never be obtained as the limit of the sequence of equilibria for finite
horizon economies.
An equilibrium in which social competition has been completely suppressed has
a source of fragility, due to the heterogeneity of agents, not exhibited in the
cooperative equilibrium of a standard prisoners’ dilemma. In our model, the
equilibrium in which competition is suppressed prescribes a punishment for a
family that deviates from the social norm – future male offspring in this line will
be matched with the poorest women of their generation. This punishment is of
arbitrarily small concern to couples at the lower end of the status distribution: their
male offspring will be matched with women who are close to the poorest even
when the norm’s prescriptions are followed.2 As a result, the punishment may be
insufficient to deter deviations by such couples. The class of equilibria in which
social competition is only suppressed among those families who get substantial
social benefits avoids the difficulty of enforcing prescribed behavior among those
elements of the society with ‘nothing to lose.’
Before going to the formal model, we should emphasize that while the focus of
this paper is on the way in which social norms can enhance efficiency, there is no
reason to believe that the norms of the sort we consider are unambiguously welfare
enhancing. Norms essentially provide incentives for individuals to behave in ways
that are not myopically optimal through the threat of future sanctions should the
norm’s prescriptions be ignored. One can easily interpret discrimination exemplified by the caste system in India or racial discrimination in the U.S. as being
1
CMP did not show uniqueness of the equilibria in which the over-saving is not suppressed.
However, there is only one equilibrium with the robustness property described below.
2
Brooks (1995) analyzes a problem with heterogeneous agents that is similarly characterized by the
difficulty in providing incentives to forgo opportunistic behavior for individuals who have little to lose
from doing so.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
9
non-myopic behavior supported by social norms. The effect of the norms that
supported these equilibria is inarguably not welfare enhancing.
2. The CMP model
There are two types of one-period-lived agents, men and women. The agents are
to be matched into pairs with each pair having two offspring, one male and one
female. In addition to the matching decision, agents make standard economic
decisions: what to consume and / or what to invest. Agents care about the economic
characteristics of potential mates.
Men and women are treated asymmetrically in two respects in order to reduce
the technical complexity of the model. First, women are endowed with a nontraded, non-storable good, while men inherit a second, storable good which we call
capital. Women are indexed by j [ [0,1] and woman j is endowed with j units of
the non-traded good. The men are indexed by i [ [0,1] and are endowed with
differing amounts of capital in the first period.
Second, only the welfare of the male offspring enters the pair’s utility function.
We normalize so that a male offspring inherits his father’s index, and we will refer
to man i, his son, his son’s son, and so on, as family line i. Consequently, parents
will make bequests only to their male offspring.
Men and women have identical utility functions defined over joint consumption
of a matched pair’s bundle given by u(c) 1 j, where c and j are, respectively, the
quantities of the male and the female goods. The utility derived from the male
good is given by a constant relative risk aversion utility function with degree of
risk aversion g ; u(c) 5 (1 2 g )21 c 12g for g ± 1, and u(c) 5 ln c for g 5 1. Finally,
the utility level of their male descendants also enters linearly into each parent’s
utility function, discounted by b [ (0,1).
The problem facing a couple is, given their wealth (determined by the bequest
from the male’s parents), how much to consume and how much to bequeath to
their son. Their son values the bequest for two distinct reasons. First, it affects the
amount he and his descendants can consume and, second, the bequest may affect
the quality of his mate. To the extent that their son’s match is affected, parents will
have an additional incentive to leave a larger bequest than they otherwise would.
Matching is voluntary with men knowing the distribution of bequests and
women’s endowments. Men prefer wealthier women (that is, those with higher
values of j), all else equal. It may be, however, that successfully matching with a
wealthy woman has adverse implications for the matching prospects of male
descendants. For example, they may be punished if their parents deviated from
prescribed behavior. Women also prefer wealthier mates, all else equal. Here, also,
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
10
matching with a wealthy man can have adverse implications for descendants’
prospects if this violates the prescriptions of the social norm.3
In each period, matches are determined by the following process: 4 each man can
make an offer to match with (propose to) any woman. Each woman then chooses
the most attractive offer if more than one proposal is received. No further offers
are made; in particular, there are no counterproposals by women to men and no
offers from men reacting to the offers of other men. If only one proposal is
received by a woman, then that proposal is accepted. If a man’s proposal is
unsuccessful, he is unmatched.5 Similarly, if a woman does not receive any
proposals, she is unmatched. A matching is a function m: [0,1] → [0,1] < h[j that
is measure-preserving and one-to-one on m([0,1]),6 where m(i) 5 j [ [0,1] is i’s
match and m(i) 5 [ indicates that i is not matched. We say that m describes a
stable matching if, taking into account future consequences, no pair would prefer
to be matched with each other rather than with their current match.
Agents use capital for current consumption and savings. Output is produced
according to
c 5 Ak 2 k9,
(1)
where k is the initial endowment capital, c is first period consumption, k9 is second
period capital, and A . 1 is a constant. To insure that the capital stock grows over
time, but family line payoffs remain bounded, we assume that 1 , Ab , Ag. We
will use interchangeably the terms output and wealth. The initial endowment of
capital for men in the first period, k 0 : [0,1] → R 1 , is a continuous and nondecreasing function of i.
2.1. The exogenous matching benchmark
An important benchmark is the case of exogenous matching. With exogenous
matching, families solve a simple decision problem with regard to their consumption and savings levels in each period. The solution to this problem describes
savings behavior when competitive over-saving to influence matching has been
suppressed by fiat.
3
Note that since wealth is passed through the sons, the male offspring from such a match may also be
wealthier (depending on the bequest decision).
4
For the case of a finite number of players, the set of subgame perfect equilibrium matchings is
precisely the set of stable matchings (Roth and Sotomayer, 1990).
5
In a model with a finite number of agents, we can simply assign the woman with the lowest
endowment to the unsuccessful man and reassign the other women, maintaining the order.
6
That is, for all measurable subsets A , [0,1], m 21 (A) is measurable and l(m 21 (A)) 5 l(A), where l
denotes Lebesgue measure.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
11
If h j t j t`50 is the sequence of matches for family line i, then family line i solves 7
O b hu(Ak 2 k
`
max
hk t j t50
t
t
t11
) 1 j t j subject to k 0 5 k 0 (i) and Ak t $ k t 11 .
(2)
This is a standard dynamic programming problem with unique solution hk t* j `t50 ,
where k 0* 5 k 0 (i), k *t 5 (1 2 l * )Ak *t 21 , and l * 5 1 2 (A12gb )1 / g. The discounted
value from consumption is V * (k) 5 ( l * Ak)12g / [(1 2 g )l * ] if g ± 1 and V * (k) 5
(1 2 b )22 h(1 2 b )ln(1 2 b ) 1 b ln b 1 ln A 1 (1 2 b )ln kj if g 5 1. With exogenous matching, there is no strategic interaction between family lines, and each
family line makes bequest decisions solely motivated by intertemporal consumption considerations.
2.2. Social norms and equilibria
A complete description of behavior (i.e., a strategy profile) in CMP is a
specification of bequest and matching behavior for each agent of all generations
after all possible histories. For a given ordering of men, a man’s status is his rank
in that ordering. We assume that a man’s status determines the quality of his
match, if all agents follow society’s prescribed rules. A man’s status (rank) can
depend upon, among other things, his own wealth and the status and match of his
parents. A social norm is a status assignment rule that determines in each
generation each man’s status as a function of the behavior of previous generations.
An equilibrium is a social norm together with a specification of matching and
market decisions such that female j matches with male of status j and no agent
wishes to deviate.
3. Single class equilibria
3.1. Wealth-is-status equilibria
An important building block of the social norms that we describe in the next
section is behavior in which social competition is not suppressed. This behavior,
7
We are assuming here, as in CMP, that the first member of the family line believes that the
descendants will make the consumption–investment decisions that that member would choose if it was
possible to dictate their choices. Since Bellman’s principle of optimality applies, we have described a
subgame perfect equilibrium of the game of consumption–investment choices with a countably infinite
number of players (one player for each generation). It is possible that there are other equilibria, which
we do not discuss.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
12
the analog to the repeated ‘not-cooperate’ equilibrium in the standard repeated
prisoner’s dilemma game (see Introduction), is the unique behavior consistent with
equilibrium if there is a finite horizon. We call the social norm that supports this
behavior the wealth-is-status social norm. 8 The characteristics of the equilibria
associated with this social norm when there is an infinite horizon are explored in
more detail in CMP. As we describe in Section 5, a crucial feature of this social
norm is its lack of sanctions on deviating individuals. Since uniqueness of the
equilibrium behavior is required to make this idea precise,9 the equilibria of the
finite horizon model and their relationship to infinite horizon equilibria are
discussed in detail in Appendix A.
To fix ideas, we first consider the two period case, in which parents are
exogenously matched in the first period.10 In the second (and last) period, matching
occurs and each man (together with spouse if he is matched) consumes his
bequest. In the first period, parents decide on the level of bequest to their sons (and
so their own consumption). Since, in the second period, women accept the
wealthiest suitor (the only consideration in women’s, and men’s, matching
decisions is current consumption), matching is necessarily positively assortative.
Thus, if F, the induced distribution of wealth in the second period by first period
bequest decisions, is strictly increasing, female j [ [0,1] is matched with the male
with wealth s satisfying j 5 F(s). It is worth emphasizing at this point that parents
can guarantee a match for their son of a woman with endowment arbitrarily close
to j by bequeathing A21 F 21 ( j) (recall (1)). The problem for couple i in the first
period is to choose the bequest k [ [0,Ak 0 (i)] to maximize
u(Ak 0 (i) 2 k) 1 b u(Ak) 1 b F(Ak).
(3)
The first order condition is 11
2 u9(Ak 0 (i) 2 k) 1 b Au9(Ak) 1 b AF9(Ak) 5 0.
Note, in particular, that the level of the bequest that solves this problem will be
strictly larger than the value of k that solves u9(Ak 0 (i) 2 k) 5 b Au9(Ak), i.e., than
the optimal bequest when matching is exogenous (in other words, in the absence
of matching considerations). This is precisely the competitive over-saving generated by the competition for spouses (of the sons by the parents) discussed above.
Except in one case, in the finite horizon model there is a unique equilibrium
associated with a given initial distribution of capital, and it is characterized by
8
As will become clear, bourgeois is perhaps a more evocative description.
The concept of an individual sanction is complicated by the existence of multiple equilibria in the
finite horizon model, since they can be used to construct punishments by ‘switching between’
continuation equilibria as a function of history (as described by Benoit and Krishna (1985) and
Friedman (1985)).
10
As will be clear shortly, this is without loss of generality.
11
We show in Appendix A that F will indeed be strictly increasing and differentiable.
9
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
13
competitive over-saving to influence matching. The exception is the case in which
the support of the initial endowment distribution of capital includes zero. In this
case, while there will be an equilibrium with over-saving, there may also exist
other equilibria. Any equilibrium in which over-saving is suppressed requires that
the distribution of individuals’ choices depend on the actions of individual agents,
which we find implausible in a model with a continuum of agents. The wealth-isstatus equilibrium of the infinite horizon economy can be obtained as the limit of a
sequence of (the plausible) equilibria of a sequence of finite horizon economies in
which the horizon goes to infinity (see Appendix A).
The wealth-is-status social norm prescribes that if a woman receives multiple
proposals, the proposal from the wealthiest man is accepted (if there is a tie, then
the woman flips a coin). Since this implies that there are no consequences from
doing otherwise, it is clearly optimal for a woman to follow the prescription. The
equilibrium matching thus depends only on the men’s capital endowments of that
period (which are determined by their parents’ bequests). In equilibrium, the
wealthiest man will match with the wealthiest woman, and so on. A man’s match
in period t depends, then, only on his relative position in the capital distribution of
period t. A wealth-is-status equilibrium is a description of consumption–bequest
decisions consistent with the wealth-is-status social norm.
To derive these consumption–bequest decisions, suppose that the bequest of
capital in period t is given by k t : [0,1] → R 1 , where k t is a strictly increasing
function. Further, suppose that in equilibrium all agents consume in such a way
that the next period capital bequest is given by k t 11 , also strictly increasing and
differentiable on (0,1), so that the match in period t 1 1 is m(i) 5 i. Consider
family line i’s choice of bequest in period t. By increasing the bequest from k t11 (i)
to an amount k in the next period, the man of index i will succeed in matching
21
with any woman such that j , k 21
t 11 (k). If we let m t 11 (k) 5 k t 11 (k) denote the
supremum of i’s match as a function of the bequest received, then the problem
facing an agent i in period t who inherits k t amount of capital is
max
0#k t 11 # Ak t
u(Ak t 2 k t 11 ) 1 m t (k t ) 1 b Wt 11 (k t 11 ) ; Wt (k t ).
(4)
If Wt 11 is differentiable, the first order condition on bequests is given by
u9(Ak t 2 k t 11 ) 5 b W 9t 11 (k t 11 ). From the envelope condition, W 9t11 (k t 11 ) 5
Au9(Ak t 11 2 k t 12 ) 1 m 9t 11 (k t 11 ). Thus, we get the following difference equation
which describes a family line’s optimal capital accumulation decision:
u9(Ak t 2 k t 11 ) 5 b hAu9(Ak t 11 2 k t 12 ) 1 m 9t 11 (k t 11 )j.
(5)
Wealth-is-status equilibria exist for all g (CMP; Cole et al., 1995b). While there
is no guarantee of uniqueness in the infinite horizon economy, there is only one
wealth-is-status equilibrium that can be approximated by the unique (robust)
equilibrium of the finite horizon games (see Appendix A). Let hk t (i)j denote family
14
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
line i’s sequence of bequests in a wealth-is-status equilibrium. Then we have the
following (from CMP):
Proposition 1. For all family lines i and all t $ 0, k t (i) $ k t* (i), with a strict
inequality for all but a set of families of measure 0.
Proposition 2. In a wealth-is-status equilibrium, if k 0 (i) . k 0 (i9), then, for all t,
the optimal level of capital in period t for agent i and i9 satisfies k t 11 (i) . k t11 (i9).
These two propositions illustrate the inefficiency of wealth-is-status equilibria.
The first proposition states that for all but a negligible set of families, the bequest
choices are higher than they would prefer had there been no competition for future
mates, while the second proposition states that the net effect of all families’
decisions cancels out. Despite the fact that all families are increasing their savings
to improve their relative position, in equilibrium, no family moves up or down in
the wealth ranking over time.
Thus, in any period every family line (except the poorest) would be strictly
better off if the identical matching were imposed exogenously, so that all families
reduce their bequest choices in that and all future periods to k *t , the levels that
solve (2). Note that this comparison only necessarily holds for the current
generation. The comparison for future generations is more delicate, since the
future cost of engaging in competitive over-saving may be outweighed by the
current benefit of receiving a larger bequest. As we pointed out in the Introduction,
the problem facing the families in any single generation is essentially a prisoners’
dilemma game: any single family has an incentive to increase its savings whether
or not other families do so, but the result of all families doing this is to make all
families worse off than if none did so.
3.2. Aristocratic equilibria
A social norm in which status was unaffected by bequest decisions would
eliminate this competitive over-saving. If such a social norm were followed,
families would no longer have an incentive to over-save. We analyzed one such
social norm, which we called the aristocratic social norm, in CMP. In the
aristocratic social norm, a man’s status is inherited; that is, his status is the same as
his father’s as long as his father matched appropriately. In the initial period, there
is an exogenously given status assignment. An aristocratic equilibrium consists of
the above status assignment rule, consumption–bequest decisions, and matching
behavior for the agents such that each man is voluntarily matched with the woman
whose endowment equals the man’s status; if a man matches with a woman with
an endowment not equal to his status, the family line from that point on has zero
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
15
status (and so all future matches are with women of zero endowment). Lastly, no
woman with positive endowment finds it profitable to match with a zero status
man when either that man’s family line has always had zero status or the status is
newly acquired.12
The aristocratic social norm described above prescribes a specific behavior and
a sanction if that behavior is violated, namely, that the violating family has its
status reduced to zero. We can imagine alternative sanctions as well, for example
ostracism. If we interpret ostracism as the complete separation from the rest of
society, ostracism would result in the extinction of the family line after one period
(unless we allowed for marriage within the family). In this view, ostracism is
essentially equivalent to death. A less extreme interpretation of ostracism is that it
does not lead to extinction of the family line, so that the male line can consume the
infinite sequence of consumption hc t j. Thus, if ostracized, a family line solves (2)
and has total discounted utility of V * (k). This is also the total discounted utility of
a family line with zero status. Moreover, since in equilibrium, no woman with
positive endowment finds it profitable to match with a zero status man when that
status is newly acquired, it is also not profitable for her to match with an ostracized
man. Thus, ostracism (in its less extreme form) is equivalent to the loss of status
specified in the aristocratic equilibrium.13 It is worth noting that this is the most
severe punishment available consistent with property rights that preclude confiscation of endowments. Of course, more severe sanctions can be constructed by
taxing or expropriating the wealth of family lines.
If everyone obeys the aristocratic social norm (i.e., the status assignment rule
and matching contingent on status), then a male from family line i will always
match with the female endowed with j 5 i, irrespective of his wealth. A family line
is punished by having its status, and hence match, set to zero forever if one of the
men in the line marries a woman other than the one prescribed by this rule.
If all families follow the prescriptions of the aristocratic social norm, bequests
will be given by k *t , the solution to (2), and so there is a zero first order cost to
increasing their bequest. Thus, in order for the aristocratic social norm to be
consistent with equilibrium, it is necessary that there be no more than a second
order gain from a small increase in bequests. In order for a deviation by family i to
be successful (in the sense of improving the quality of the match of their son),
three deviations are necessary. First, the parents must make a bequest above k *t (i)
(otherwise their son’s match would be j 5 i even if women accepted the wealthiest
suitor). Second, the son must make a proposal to a higher ranked woman j . i and,
12
We are only requiring that it be optimal for a woman to follow status on the equilibrium path and
for one generation off the equilibrium path. This issue is discussed in detail in CMP.
13
This equivalence does not hold if the distribution of women’s endowments is bounded away from
zero.
16
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
finally, the offer must be accepted.14 The trade-off facing the parents in family line
i is whether the one period increase in quality of match is sufficient to outweigh
the permanent future loss of status (valued at b 2 i /(1 2 b )) as well as the one
period distortion in bequest behavior (that is of zero first order cost). The trade-off
for the son is similar, although the loss of status occurs one period earlier and there
is no distortion of bequest behavior. Finally, the trade-off for the potential spouse
is the comparison between the increase in consumption and loss of status. Recall
that, by assumption, if the potential spouse receives a higher utility (including
future costs or benefits) from the offer of the deviating male, then she will accept
it.
If both family line i and female j (and consequently j’s son) have low status,
then the loss of status is not a severe threat, which makes the existence of
aristocratic equilibria delicate. For example, if k 0 (0) . 0, then aristocratic equilibria cannot exist. To see this, let Lt ( j) be the capital level that leaves the woman
indifferent between matching with the man of status j with capital k t ( j) and
maintaining the current status for her son and matching with a man with capital
Lt ( j) and having her son receive zero status. If the woman does match with the
latter, the status of the man that the woman deviates to is irrelevant since her
offspring will have zero status. Thus, the capital level Lt ( j) is described by
j
V * (Lt ( j)) 1 j 5V * (k t ( j)) 1 ]].
12b
Note that Lt ( j) exceeds k t ( j) by an amount that compensates for the value of lost
status. It is worth noting here that Lt ( j) is close to k t ( j) for j close to zero.
Consider now the bequest decision of family line i in period t 2 1. Bequeathing
Lt ( j) will improve their son’s match, but at a cost of reduced consumption today
as well as lower status after two generations. Bequeathing k t (i) yields a (weakly)
higher payoff than Lt ( j) if
V * (k t 21 (i)) 1 i /(1 2 b ) $ u(Ak t 21 (i) 2 Lt ( j)) 1 i 1 b j 1 bV * (Lt ( j)).
(6)
This inequality holds with equality when i 5 j 5 0. While the left hand side is
independent of j, the right hand side is strictly increasing in j (its derivative with
14
The necessity of the first deviation follows from our assumption that the original assignment of
status is that family line i has status j 5 i, so that if bequest behavior and matching is in accordance
with the candidate equilibrium, then matching will be positively assortative. If aristocratic equilibria
exist, additional aristocratic-type equilibria can be constructed in which the status assignment deviates
somewhat (particularly at higher wealth levels) from the relative position in the initial wealth
distribution. In these other aristocratic equilibria, family i may be successful in a deviation after only
two deviations (if the son already has wealth greater than that of the intended match for some j . i).
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
17
respect to j is b 1 ( b (V * )9 2 u9)L t9 5 b . 0 at i 5 j 5 0). Thus, the inequality is
violated for i and j small, and aristocratic equilibria do not exist when k 0 (0) . 0.15
The crucial feature is that, when k 0 (0) . 0, the family line with zero status can
attract a significantly better mate for their son by bequeathing enough. Similarly,
all family lines with sufficiently low status can also attract significantly better
mates for their sons than they would by following the prescription of the social
norm. Low status families gain little by following the social norm and, consequently, have little to lose if they deviate from its prescriptions. When k 0 (0) . 0,
these families have the wealth to deviate from the norm’s prescriptions in ways
that lead to net benefits.
Aristocratic equilibria may exist when k 0 (0) 5 0. In this case, low status family
lines cannot attract significantly better mates for their sons, because low status
family lines simply do not have sufficient wealth to induce a better mate to deviate
from her prescribed match. Phrased differently, any family with sufficient wealth
to induce a deviation by a significantly better quality match has sufficient status
that its loss will deter a deviation. However, even when k 0 (0) 5 0, the question of
the existence of aristocratic equilibria is delicate.16 In Cole et al. (1995b), we
provide a sufficient condition for the existence of aristocratic equilibria, when
u(c) 5 ln c and k 0 (i) 5 k i, for some k . 0.17
In summary, while wealth-is-status equilibria exist under standard regularity
conditions, aristocratic equilibria exist only under quite restrictive conditions.
Eliminating the incentive to over-save is impossible in most circumstances.
However, while a social norm may not be able to eliminate the incentive to
over-save, it can ameliorate the incentive to over-save by dividing society into
groups, solving the problem in some but not all groups. The division of society
into groups which differ in the degree to which social competition is controlled
also introduces an additional control instrument: the threat of reassigning deviators
in a group to another group in which the competition has not been controlled.
15
More generally, suppose that the value of a match with female j is v( j) (rather than j). The above
argument applies as long as v9(0) . 0. Aristocratic equilibria will exist when k 0 (0) . 0 for some
specifications of v (this would, of course, require that v9(0) 5 0).
16
The condition in CMP for the existence of aristocratic equilibria was subsequently shown to be
vacuous by Landsburg (1995).
17
Landsburg (1995) provides a condition (condition 1) that avoids the nonexistence problem by
guaranteeing a mass of males who have no wealth and consequently cannot deviate from the social
norm. This creates a uniform, bounded away from zero per period punishment that can be imposed on
all potential deviators with a nontrivial decision. Furthermore, since the punishment entails that all
future males in this line will be affected in this way, the cost of this punishment can be made arbitrarily
large by choosing a discount factor sufficiently close to 1. This argument has the flavor of the folk
theorem in repeated games and relies on the existence of a sufficiently large punishment that can be
imposed on deviators to ensure concurrence with a prescribed equilibrium.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
18
4. Equilibria with class systems
In this section we explore social arrangements that ameliorate inefficient savings
competition. The social norms we examine divide society into groups, attempting
to restrict competition in some groups, but not all. Individuals in groups in which
the competition is eliminated will face the threat of being ejected from the group if
they deviate from the norm’s prescriptions. We interpret the partition of society
into different groups as constituting different social classes when there is a
qualitatively different determination of status between the groups.
The first social norm we study, in Section 4.1, is an aristocratic two-class
equilibrium: the population is divided into two groups, with the bottom group
governed by the wealth-is-status social norm and the upper group governed by the
aristocratic social norm, modified so that if there is a deviating match, the resulting
family line is relegated to the lower group. Hence, over-saving is eliminated in the
upper class, but not in the lower class. The question of existence of an equilibrium
consistent with this social norm is delicate; we provide necessary and sufficient
conditions for existence only in terms of the endogenous consumption value
functions (Proposition 3).
Section 4.2 describes and gives sufficient conditions on exogenous parameters
(with no assumptions on the endogenous consumption value functions) for the
existence of a variant of this two-class social norm. It differs from the aristocratic
two-class social norm in that the matching is random in the upper class. Even
when the aristocratic two-class social norm does not exist, the variant social norm
eliminates over-saving in the upper group.
Finally, in Section 4.3, we combine elements of the previous two social norms
to describe a three-class social norm, in which the bottom class follows wealth-isstatus, the middle class follows random matching but is not over-saving, and the
upper class is aristocratic and is not over-saving. The conditions for existence of
the three-class equilibrium are similar to that for the randomized two-class
equilibrium.
4.1. An aristocratic two-class equilibrium
We define an aristocratic two-class status social norm as follows. Given a
designated index il [ (0,1), we divide both men and women into two groups, [0,il ]
and (il ,1], called, respectively, the lower group and the upper group.18 Status for
18
It simplifies discussion to place il in the lower group. Nothing substantive hinges on the choice of
group that il belongs to.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
19
men in the lower group is determined by wealth, and matching in this group is as
in the wealth-is-status social norm: the woman with index i marries the man with
status i. Male i in the upper group has status i (and so matches with female j 5 i)
as long as the men in that family line have followed the prescribed matching. If
there is a deviation by a man or a woman in the upper group, then male
descendants are assigned thereafter to the lower group.
Let hkˆ t (i)j `t 51 be the sequence of bequests by family line i [ [0,1] in the
wealth-is-status equilibrium and hVˆt (kˆ t (i))j be the sequence of values of the
discounted utility streams from consumption for family line i. If the women in the
upper group reject all offers from men in the lower group, the men and women in
the lower group behave as a separate society following the wealth-is-status social
norm. Since the savings behavior of family lines i , il is independent of the value
of il ,19 the savings behavior of the lower group is given by hhkˆ t (i)j `t 51 : i [ [0,il ]j.20
The existence of this lower group, with its inefficient savings decisions, creates the
possibility that the simple threat of relegation to the lower group may be sufficient
to deter deviation.
Initial wealth, k 0 , is nondecreasing in index; hence, agents in the upper group
(the ‘aristocrats’) are initially at least as wealthy as agents in the lower group.
Thus, in the first period, Vˆ1 (kˆ 1 (i)) # Vˆ1 (kˆ 1 (il )) ,V * (k *1 (i9)) for all i # il # i9, and
so there is a loss of consumption utility from being relegated to the lower group
(and for this reason, there is also no possible gain from deviating for a woman in
the upper group). However, the lower group family lines are accumulating wealth
at a faster rate than the upper group family lines, and k 0 is continuous, so some
lower group men will be wealthier than some men in the upper group in the next
period; i.e., kˆ 2 (il ) . inf i 9oi l k 2* (i9) 5 k 2* (il ). It is not clear how Vˆt (kˆ t (il )) and
V * (k t* (i9)) rank for t $ 2 and i9 . il . It is possible that some lower group family
lines asymptotically become extremely wealthy relative to family lines in the
upper group.
A necessary condition for the aristocratic two-class social norm to be consistent
with equilibrium is that
19
The level of bequests by family line i is determined by the need to bequeath more than the family
lines below i. Specifically, the bequests are determined by an initial value problem with an initial value
at 0 (see Appendix A).
20
As we discussed in Section 3.1, there may be more than one wealth-is-status equilibrium of the
infinite horizon model. For any such wealth-is-status equilibrium of the lower group (considered as a
separate society), there is a wealth-is-status equilibrium of the entire society that agrees with it on the
lower group and, conversely, every wealth-is-status equilibrium of the entire society also describes a
wealth-is-status equilibrium of the lower group considered as a separate society. However, even if there
are multiple equilibria, only one is the limit of finite horizon games (see Appendix A), and this
equilibrium, restricted to the lower group, has the same property.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
20
Vˆt (kˆ t (il )) #V * (k t* (il )) for all t.
(7)
To see this, suppose that Vˆt (kˆ t (il )) .V * (k t* (il )) for some t. Then, for i9 close to il ,
Vˆt (kˆ t (il )) .V * (k t* (i9)). A deviating offer by male il to a female i9 . il is profitable
for male il if accepted. But the deviating offer is accepted if
b (i9 2 il )
Vˆt (kˆ t (il )) 2V * (k t* (i9)) . ]]],
(1 2 b )
which holds for i9 close to il .
Moreover, (7) is almost (but not quite) sufficient. Certainly, if Vˆt (kˆ t (il )) #
V * (k t* (il )) for all t, then no female in the upper group will accept an offer of
k # kˆ t (il ) from a male in the lower group – it involves both a loss of status and a
reduction in family line consumption utility. However, we must also rule out the
possibility that, for example, the parents in family line il in period t bequeath
k . kˆ t (il ) in order to offer a consumption utility of strictly more than Vˆt (kˆ t (il )).21
In order to see why (7) may not be sufficient, suppose Vˆt (kˆ t (il )) 5V * (k t* (il )) in
period t. Generation t 2 1 of family line il , by bequeathing strictly more than kˆ t (il )
to their son, can guarantee him a better match than il . The increase in the index of
the female that male il can induce to deviate, however, is of the same order as the
increase in the bequest over kˆ t (il ). A similar (but not identical) trade-off faces
family line il in the wealth-is-status social norm. Note, in particular, that in the
wealth-is-status equilibrium, it is not optimal for generation t 2 1 to increase their
bequest from kˆ t (il ) to any k . kˆ t (il ), even though their son’s spouse’s index
increases from il to (kˆ t )21 (k).22 Let i(k) be the highest index spouse from the upper
group that il can induce to deviate after receiving a bequest of k. Then, for k close
to kˆ t (il ) (so that it is optimal to consume all of Ak 2 kˆ t 11 (il ) in one period), i(k)
satisfies
b il
i(k)
u(Ak 2 kˆ t 11 (il )) 1 i(k) 1 b Vˆt 11 (kˆ t 11 (il )) 1 ]] 5V * (k *t (i(k))) 1 ]].
12b
12b
(8)
By construction, generation t 2 1 is indifferent between the bequest-match pair
(kˆ t (il ),il ) and the pair (k,(kˆ t )21 (k)), where k is arbitrarily close to kˆ t (il ). Thus, if
i(k) . (kˆ t )21 (k) for k larger than, but in a neighborhood of, kˆ t (il ), then generation
21
If any family line in the lower group has a profitable deviation, then so does family line il .
Recall that kˆ t (i) is the level of wealth in period t of family line for i [ [0,1] in the wealth-is-status
equilibrium.
22
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
21
t 2 1 will deviate. While we have been unable to verify that this inequality holds,
the structure of the wealth-is-status equilibrium suggests that it does.23
However, at least for the case g . 1, a slight strengthening of (7) is sufficient.
The sufficient condition is: there exists d . 0 such that, for all b large and all t,
Vˆt (kˆ t (il )) 1 d #V * (k t* (il )). Given such a uniform lower bound on the gap between
the discounted stream of consumption utility in the upper and lower groups, any
offer acceptable to a female in the upper group must involve a wealth level
sufficiently higher than kˆ t (il ). Any such offer involves a sacrifice by the parents in
family line il (of utility order d ) that can only be outweighed by the upper group
female having an index bounded away from il . Denote this bound by h . 0. Thus,
it is enough to deter females with index j [ (il 1 h,1], and any female in the upper
group with an index j $ il 1 h suffers a status loss of at least h per period. It is
here that the size of g is relevant: if g . 1, the consumption utility function is
bounded above (by zero), and so there is a bound on how much additional utility
can be offered to a potential deviator through increased consumption. Once people
are sufficiently wealthy, the additional utility from any increase in consumption
will be dominated by the loss of status, bh per period (recall that the loss of status
impacts on the match of the son and so is discounted by one period). This
immediately allows us to conclude that, if all the men in the upper group are
sufficiently wealthy, then no woman in the upper group will be willing to deviate
from the prescribed behavior. Since the initial wealth is positive, eventually all the
agents in the upper group become sufficiently wealthy, and, as the proof of
Proposition 3 below demonstrates, for sufficiently high discount factors, the gain
23
21
Since i(kˆ t (il )) 5 (kˆ t ) kˆ t (il ) 5 il , if
d(kˆ t )21
di(k)
] k 5kˆ (i ) .]] k 5kˆ (i ) ,
dk
t l
dk
t l
U
U
then the inequality holds. Now,
( b 1(12 b )dV * / dk?dk * / di)di(k) / dk5(12 b )Au9(Ak2kˆ t 11 (il )),
21
while (5) implies (since m t (k) 5 kˆ t (k) and k 5 kˆ t (i) in equilibrium) that
21
d(kˆ t )
]]5 b 21 (u9(Akˆ t 21 (i)2kˆ t (i))2 b Au9(Akˆ t (i)2kˆ t 11 (i))).
dk
21
For sufficiently large wealth, di(k) / dk and b (1 2 b )Au9(Ak 2 kˆ t 11 (il )) have almost the same slope.
Thus, the inequality holds if Au9(cˆ t ) . u9(cˆ t 21 ), where cˆ t 5 Akˆ t (i) 2 kˆ t 11 (i). For g . 1, this is
equivalent to A1 / g . cˆ t /cˆ t 21 . This condition will be met if cˆ t is not too large relative to cˆ t 21 . Since
consumption utility is bounded, for sufficiently large wealth, once a critical level of consumption in
each period is achieved, most of the further increases in wealth will be reinvested (because of the
competition for mates) rather than consumed [see CMP (Property 6)], suggesting that consumption may
not grow very quickly.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
22
from increased consumption before agents become sufficiently wealthy is outweighed by the loss of status.
Finally, we consider the possibility of deviations by males in the upper group.
The considerations that yield nonexistence of aristocratic equilibria when k 0 (0) . 0
do not arise here. The argument from the previous paragraph can be used to show
that, for sufficiently high b, males in the upper group have no profitable
deviations. In particular, consider generation t 2 1 of family line il and suppose
(momentarily) that family line il is a member of the upper class. As we have just
seen, under the sufficiency condition described in the previous paragraph, a
marginal increase in their bequest (which incurs only a second order loss) does not
allow their son to secure a marginally better mate (in contrast to the aristocratic
social norm with k 0 (0) . 0).
Once we take these into account we have (the proof is in Appendix B): 24
Proposition 3. A necessary condition for the aristocratic two-class social norm to
be consistent with equilibrium is (7):
Vˆt (kˆ t (il )) #V * (k t* (il )), ;t.
If g . 1 and there exist d . 0 and b¯ such that
Vˆt (kˆ t (il )) 1 d #V * (k t* (il )), ;t
(9)
holds for b [ ( b¯ ,1), then there exists b¯ . b¯ such that the two-class aristocratic
social norm is consistent with equilibrium for b [ ( b¯ ,1).
If two-class aristocratic status social norms are consistent with equilibrium, all
family lines are at least as well off (in terms of period zero discounted utilities) in
the two-class aristocratic social norm as in the wealth-is-status social norm. Those
agents in the lower group receive the same outcome, and so utility, as in the
wealth-is-status social norm, while agents in the upper group receive the same
matches, but enjoy higher consumption utility.
The argument above only demonstrates that all family lines are at least as well
off under the two-class norm as under the wealth-is-status norm. Since the upper
class saves less in the two-class equilibrium than in the wealth-is-status equilibrium, future generations of families in the upper class will be poorer in the
two-class equilibrium than they would be in the wealth-is-status equilibrium.
Hence, the argument above does not say anything about the welfare of these future
generations. However, from Proposition 3, at least for family line il , the welfare
comparison is true in every generation, not just the initial one. Thus, if the initial
wealth distribution has k 0 (i) 5 k¯ for all i $ il , then all generations of all family
24
Note that this proposition (unlike the other propositions) does not provide conditions on exogenous
parameters that guarantee existence of aristocratic two-class equilibria.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
23
lines are at least as well off in the two-class aristocratic social norm as in the
wealth-is-status social norm.
4.2. A randomized two-class equilibrium
Even if Vˆt (kˆ t (il )) #V * (k *t (il )) for all t fails for b close to 1, it is still possible to
have a two-class equilibrium. However, in order to deter deviations by agents with
low aristocratic status, those at the bottom need to be given additional incentive to
follow the social norm. This can be done by using a random matching rule in the
upper group.
We define a randomized two-class status social norm as follows. As before,
given a designated index il [ (0,1), men and women are divided into two groups, a
lower group [0,il ], and an upper group (il ,1]. Status for men in the lower group is
determined by wealth while all males in the upper group have identical status (as
long as the men in that family line have followed the prescribed matching and
consumption patterns). For men and women in the upper group, matching is
random; that is, any man in (il ,1] is equally likely to be matched with any woman
in (il ,1].
Since the random matching within the upper group may match a high index
woman with a man just above il , it is not immediate that men in the upper group
will not make deviating offers to women in the upper group. Deviations from
prescribed behavior for either men or women in the upper group result in their
being put into the lower group, as will be all descendants in the line. In essence, a
deviation from the prescribed rules in the upper group results in the family line
being banished from the upper group. Since the expected value of a match for any
male in the upper group is (1 1 il ) / 2, being dropped into the lower group results in
a loss of at least (1 2 il ) / 2, since the best match possible if one is in the lower
group yields il . Since the discounted stream of utility from consumption is
maximized in the exogenous matching case, it is easy to deter men in the upper
group from deviating. The maximum possible gain from deviating is improving
the current match by no more than 1 2 il . The discounted value of the loss of
status from the next period is b (1 2 il ) /(2(1 2 b )), with an additional cost coming
from the need to distort consumption–savings decisions from the exogenous
matching case. If b $ 2 / 3, just the cost of loss of status is sufficient to deter males
in the upper group.
There is also, however, as in the aristocratic two-class social norm, the
possibility that men in the lower group may attempt to match with upper group
women. As before, the lower group family lines are accumulating wealth at a
faster rate than the upper group family lines. In order for the two-class social norm
to be consistent with equilibrium, it is necessary that upper group women reject
very wealthy suitors from the lower group. If the utility from consumption is
unbounded (g , 1), the loss of status may not be a sufficient cost to ensure that
upper group women reject these suitors.
24
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
When g . 1, the consumption utility function is bounded above (by zero). Then,
once people are sufficiently wealthy, the additional utility from any increase in
consumption will be dominated by the loss of status, b (1 2 il ) / 2. Here, as in
Proposition 3, this allows us to conclude that, first, if all the men in the upper
group are sufficiently wealthy, then no woman in the upper group will deviate and,
second, for sufficiently high discount factors, the gain from increased consumption
before agents become sufficiently wealthy is outweighed by the loss of status.
Suppose now that the man with index il and all men above him are governed by
the status assignment rule described above while men below are governed by the
wealth-is-status rule and that k(i) . k(il ) for all i $ il . Given that all agents with
indices greater than or equal to il will not deviate from the social norm under any
circumstances, those agents below can be treated as a society by themselves;
nothing they do will affect the behavior of the agents in the upper group, and
nothing the upper group does will impact on the lower group. But there will exist a
wealth-is-status equilibrium for the agents in the lower group, considered as a
distinct society. If we concatenate the wealth-is-status equilibrium for the lower
group with the randomized social norm for the upper group, this will constitute a
two-class equilibrium. This motivates our next proposition (the proof of which
mimics the first part of the proof of Proposition 3, with j 2 il 1 h replaced by
(1 2 il ) / 2):
Proposition 4. Suppose g . 1. Fix il [ (0,1) such that k 0 (il ) . 0. There exists
bˆ , 1 such that for b . bˆ , the randomized two-class social norm with upper
group (il ,1] is consistent with equilibrium.
We now compare the utility levels people would have in the first period in the
wealth-is-status equilibrium and the randomized two-class equilibrium. For the
agents at the bottom, there is no difference since they are in a wealth-is-status
equilibrium relative to the bottom group in either case. The agents at the top are
not indifferent, however. Those above the average for the upper group, (il 1 1) / 2,
are receiving lower expected value from their match in the randomized two-class
equilibrium than in the wealth-is-status equilibrium while those below the average
are receiving higher expected utility from the matches. All agents in the upper
group are better off with respect to utility from consumption in the randomized
two-class equilibrium since they no longer have the distortion that results from the
competition in the wealth-is-status equilibrium. Thus, those males below (il 1 1) / 2
are clearly better off in the randomized two-class equilibrium since they are better
off with respect to both matching and consumption.
What about the welfare of males above (il 1 1) / 2? For the man with index 1,
his loss in utility from matching is (1 2 il ) / 2. But it is possible that this loss is
greater than the utility increase from consumption. In that case, this individual is
strictly worse off in the randomized two-class equilibrium than in the wealth-is-
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
25
status equilibrium. We do not know if two-class social norms that necessarily
Pareto dominate the wealth-is-status social norm exist.
As a final, more speculative, comment, we note that the two-class equilibrium of
Section 4.1 makes finer distinctions between the agents than the randomized
two-class equilibrium in this section. As we saw, randomized two-class equilibria
appear to be less fragile than non-randomized two-class equilibria because they do
not treat different upper-class family lines differently. Is this an explanation for
why many class systems appear to avoid too fine a distinction between members?
4.3. A three-class equilibrium
Building on two-class equilibria, we can also construct equilibria with low,
middle, and upper classes. The idea is that aristocratic matching can be supported
in the upper class since the middle class has random matching. Fix il , i0, and
define a three-class status assignment rule as follows: status for men with index
i # il is determined by wealth-is-status, all men with index i between il and i0 have
the same status (any man in (il ,i0] is equally likely to be matched with any woman
in (il ,i0]), and a man with index i . i0 has status j 5 i. Deviations from prescribed
behavior for either men or women in either the middle or the upper class result in
their being put into the class just below, as will be all descendants in the line.
In the three-class equilibria, the (one period) value of loss of status in moving
from one class to the one below is at least (i0 2 il ) / 2, since the expected value of a
match for any male in the middle class is (i0 1 il ) / 2, the infimum of status in the
upper class is i0, and the supremum of status in the lower class is il . A similar
argument to that showing the existence of two-class equilibria shows that threeclass equilibria exist, and so we have:
Proposition 5. Suppose g . 1. Fix il , i0 [ (0,1), and suppose 0 , k 0 (il ) # k 0 (i0).
There exists bˆ , 1 such that if b . bˆ , there exists a three-class equilibrium in
which men with index i . i0 are governed by the aristocratic status assignment
rule, those with index i [ (il ,i0] are governed by the random status assignment
rule, and those with index i # il are governed by the wealth-is-status rule.
5. Discussion
We believe the equilibria we have constructed capture salient features of the
class system: behavioral prescriptions differ by class, and individuals match within
their class. The model makes concrete predictions about the extent to which and
the manner in which social competition can be suppressed with a class system. A
central feature of the analysis is that social competition is not suppressed in the
lowest class. Further, the poorest family line within the class just above the poorest
must enjoy a discrete advantage from being in this class if it is to be prevented
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
26
from deviating. This is accomplished by the random matching rule in the
randomized two-class social norm.
In our model, competition takes the form of over-saving. One attractive aspect
of this form of competition is that it captures the intertemporal problem associated
with seeking to restrain social competition: individuals can seek to improve the
position of their family line over time and thus be able to undercut social
prescriptions that determine their status position. This form of competition is
constructive for the individuals involved since it raises their offspring’s future
consumption and matching opportunities, but destructive from a social perspective
since the increase in consumption is not sufficient to offset the welfare loss to the
family line of forgone consumption today. While the ability to affect their future
matching opportunities has a zero-sum aspect socially, the increase in wealth does
enable an individual to offer his mate more consumption, hence making her better
off. Purely destructive competition could arise if competition were entirely for
signalling purpose. (For a discussion of conspicuous consumption as a signalling
device see Cole et al. (1995a).)
While we have focused on social norm equilibria that ameliorate self-interested
behavior, our analysis provides insights about the structure and existence (or
nonexistence) of equilibria in a broad range of models with heterogenous agents.
In particular, the social norms that do not limit self-interested behavior do not
require, implicitly or explicitly, sanctions to be imposed on deviating individuals.
The equilibrium behavior is strategically similar to behavior that is sometimes
called myopic or anonymous. For example, if the interaction between members of
the community can be represented by repeated play of a constant stage game, then
myopic behavior involves choosing a stage-game best reply to the behavior of
other members. For communities with a large number of members, anonymous
equilibria specify that agents react only to some aggregate statistic of behavior,
disregarding the identities of agents choosing different actions; moreover, a single
agent typically cannot affect the aggregate statistic. This is also related to the
notion of a Markov perfect equilibrium, in which behavior only depends upon
payoff-relevant aspects of history. However, multi-class social norms may allow
us to construct equilibria with lower payoffs than social norms that do not involve
sanctions.
We have used the term class in a very particular way that is not the only
reasonable way to use the term. Burdett and Coles (1997) analyze a dynamic
model in which there are continua of men and women who engage in costly search
for mates. As in our model, both men and women are ranked according to
desirability, with preferences over potential mates being identical for all members
of each gender. Whereas we analyze the case in which matching is frictionless,
they treat the case of costly search. They show that under quite general
circumstances, the equilibria of their model exhibit a particular structure that they
term class structure.25 In this structure, both the set of men and the set of women
25
Burdett and Coles (1997) discuss other work with similar analyses.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
27
can be divided into ranked groups, in which the most desirable men are in the top
group, the most desirable of the men not in the top group are in the second group,
and so on. The women are similarly grouped. In equilibrium, the matching takes
place only within groups; that is, the men of the top group match only with women
in the top group, men in the second ranked group match only with women in the
second ranked group, etc. Burdett and Coles (1997) use the term class to describe
a group that thus arises endogenously in equilibrium.
The use of the term class to describe a qualitative feature of the equilibrium of
their model distinguishes their work from ours. As discussed in the Introduction,
our aim in this paper is to present a notion of class that went beyond describing
qualitative features of an equilibrium to encompass limits on agents’ opportunities
beyond those imposed by strict economic circumstances. In the language of our
paper, their equilibrium constitutes a single class equilibrium with the only
differences in the alternatives available to any agents being those resulting from
differences in their endowments.
A feature of the way we have used class in this paper is that there is some
exclusivity. Agents are not anonymous; put another way, there will be situations in
which agents with identical economic characteristics have different opportunity
sets. This feature is related to our desire that class be more than a terminological
device to describe an equilibrium. Clearly, if membership in a class is not
determined solely on the basis of economically relevant characteristics and
membership in a class affects opportunities, the consequence is that there is some
exclusivity: opportunities are not determined by exogenous economic characteristics.
We can use the model presented to think about the class structures of different
countries. The logic of the model suggests how an equilibrium in which both an
aristocratic and a bourgeois social norm were being followed could be destabilized. For a norm to be followed in equilibrium in our model, the possible gains
from deviating from the norm’s prescribed behavior are smaller than the losses
associated with the ostracism from one’s class that will result from the deviation.
Two things can destabilize the equilibrium: an increase in the value of consumption a man in the lower class can offer an aristocratic woman or a decrease in the
losses associated with being cast out of the aristocratic class to the lower class.
Shocks that result in a high marginal utility of consumption for members of the
aristocratic class can make deviations from the norm profitable. A plausible
conjecture is that the decline in the traditional aristocracy in England arose from
just such behavior. Novelists such as Jane Austen made one of the basic themes in
the English novel the issue of money versus status as a decisive factor in marriage.
We can think more generally about the determination of the costs of ostracism.
For simplicity, our model treats a particular group of people as though they existed
in isolation. Ostracism from the upper class in our model results in automatic
placement into the lower class. In reality, the norms that affect the behavior of
people are determined not among all the people in the world, but in smaller
subgroups. Depending on the particular question, the natural subgroup might be all
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
28
the people in a single country, those in a particular village in that country, or the
members of a church in that village. An individual who is barred from economic
and social participation in one class within a group will be forced to seek
interaction elsewhere. It is the relative desirability (or undesirability) of the
‘outside options’ that governs the limits on behavior that can be enforced within a
norm equilibrium. Behavioral constraints on individuals in an isolated Indian
village will loosen substantially when inexpensive bus service to distant cities
becomes available; deviation from the norm results in an association with a less
attractive set of people rather than banishment to solitude. It is the extremely high
cost of leaving the closed communities associated with certain religious groups
such as the Amish that enables these groups to enforce such atypical behavior.
These observations are not completely novel. However, the advantage of a fully
specified model such as ours in examining such questions is that it provides
specific and potentially testable implications. The two-class equilibrium analyzed
exhibits behavior on the part of the upper class that must be supported by the
threat of future sanctions imposed on the descendants of deviators. This equilibrium behavior is similar to that in repeated games and it is generally understood
that such enforced cooperation is more difficult as the group cooperating is larger
or more heterogenous. In principle, one could examine whether these variables are
related to the breakdown of aristocratic norms.
Similarly, a rapid rise in the wealth of members of the lower class could cause a
breakdown of the aristocratic norm. A number of people have suggested that the
rapid increase in the number of men in England of non-aristocratic birth who
accumulated fortunes during the industrial revolution lead to a decline in the
aristocracy in that country.26 If a rise in the number of rich, lower class suitors
increases the cost of following the aristocratic social norm, a decrease in the
well-being of the lower class increases the cost of deviating from it. One would
expect to see decreases in the standard of living for those outside the upper class to
be accompanied by a closer adherence to the aristocratic social norms by those in
the upper class.
The particular form through which the existence of a lower class increases the
possibility of cooperation that is inconsistent with self-interested behavior is
reminiscent of efficiency wage models (these are surveyed in Weiss (1990)). In
these models, workers have the opportunity to shirk on the job, with there being a
positive probability that the shirking will not be detected. The shirking, while
advantageous to the worker, is inefficient in that the cost to the employer of
shirking is higher than the benefit to the worker. Efficiency wage models
demonstrate how unemployment, in equilibrium, can serve as a deterrent to
shirking. Workers can be paid above their reservation wage, giving them a strict
gain from employment as compared to unemployment. If there is no unemployment (or in other words, if workers can be sure of being immediately reemployed
26
See, for example, Stone and Stone (1984).
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
29
after losing their job), the threat of being fired imposes no cost. If, however, there
is unemployment in equilibrium, a worker who is fired will not be sure of being
immediately reemployed and, consequently, will be less likely to shirk. The
efficiency improvements associated with decreased shirking cannot be obtained if
there is full employment. The existence of an unemployed group in conjunction
with a strict preference for being employed gives workers something to lose if they
are caught shirking and consequently decreases the incentive to shirk. The parallel
between the group of unemployed workers and the lowest class in our model is
clear: the existence of each provides an inducement for those outside the groups to
cooperate in order to stay out of these groups.
There is an important distinction, however, between our work and efficiency
wage models. A central assumption in those models is anonymity. If all firms
know the history of each worker, then any worker caught shirking can be punished
by being paid only his or her shirking reservation wage thereafter by every firm
that hires that worker. In contrast, wealth-is-status equilibria are the only
anonymous equilibria in our model.
6. Conclusion
Our goal was to construct a model in which class structures can affect behavior,
with the composition of the classes affected by economic incentives. More
specifically, we wanted to illustrate how multiple classes might exist, with the least
constraining norms (wealth-is-status in our example) necessarily applying to those
who benefited the least from the social benefits (matched with low index women in
our example), while the more constraining norms applied to those classes for
which membership yielded greater social benefits. The model presented above
fulfils this goal, but obviously it is only one of many that would do so. Many of
the details of the model were chosen for analytic tractability or pedagogical
simplicity. Any model that adequately describes the interaction of class structures
and economic incentives must have both economic and social components. We
modeled pairs of agents as receiving utility from both consumption and the quality
of their offspring’s match; we could have chosen other mechanisms to generate
social benefits. Having chosen matching as the social benefit, we modeled the
trade-off with economic concerns through a particularly simple additively-separable utility function. While this simplifies analysis, it results in bounded social
benefits that are independent of the economic circumstances. Consequently, if over
time the utility from consumption goes to infinity, social concerns are asymptotically swamped by the economic component. Since we strongly believe that
social concerns are not asymptotically irrelevant, we used a parametric form for
the utility function that bounded consumption utility, thereby ensuring that there
always will be (even asymptotically) a nontrivial trade-off between the social and
economic components. A by-product of this formulation is that, in the two-class
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
30
equilibrium, members of the lower class asymptotically become wealthier than
members of the upper class. We do not view this as a testable prediction of our
model, but rather as a consequence of the choice of the particular parametric form
utility function. A possibly more plausible (but less tractable) choice of utility
function that (we believe) does not exhibit this feature is one in which the utility
function is unbounded in consumption and the social concern enters multiplicatively. Our model is meant to illustrate an essential interdependence between social
and economic concerns and how that interdependence can support class systems; it
is not meant to be a ‘structural’ model that can be ‘tested’ through an investigation
of the individual details comprising the model.
Acknowledgements
This paper has benefited from the comments of the participants of a number of
seminars. Mailath and Postlewaite gratefully acknowledge support from the
National Science Foundation. Mailath thanks the Australian National University
for its gracious hospitality while this version was written.
Appendix A
The finite horizon CMP model
In this appendix, we analyze the CMP model with a finite horizon: the nth
generation is the last generation, with the output generated by the bequest from
generation n 2 1 completely consumed by the son and his mate and the game
ending at that point. In the last period, if a woman receives multiple offers, then
she will accept the offer from the wealthiest man (since this completely determines
her utility). The two period case, with a continuous and strictly increasing initial
capital endowment k 0 , was introduced in Section 3.1.27 An equilibrium is described
by a function s : [0,1] → R 1 , where s (i) is the bequest of family i. Recall that
women accept the wealthiest suitor, so that matching is necessarily positively
assortative, and the problem for couple i in period 1 is to choose the bequest
k [ [0,Ak 0 (i)] to maximize
u(Ak 0 (i) 2 k) 1 b u(Ak) 1 b F(Ak).
27
(A.1)
A two period example in which all parents have the same initial capital endowment is discussed in
Section IV.A of CMP.
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
31
If s is a strictly increasing bequest function followed by the other agents,28 then
F(Ak) 5 lhi: As (i) # Akj 5 lhi: i # s 21 (k)j 5 s 21 (k), where l is Lebesgue measure. The first order condition is
1
2 u9(Ak 0 (i) 2 k) 1 b Au9(Ak) 1 b ]]]
5 0.
s 9(s 21 (k))
In equilibrium, k 5 s (i), so if s is a strictly increasing equilibrium bequest
function, it satisfies the differential equation:
b
s 9(i) 5 ]]]]]]]]].
u9(Ak 0 (i) 2 s (i)) 2 b Au9(As (i))
(A.2)
Note, in particular, that this requires that s (i) . k ** (i), where k ** (i) solves
u9(Ak 0 (i) 2 s (i)) 5 b Au9(As (i)); i.e., k ** is the optimal bequest in the absence of
matching considerations.
Suppose that k 0 (0) . 0. Since the poorest family has a positive amount of
wealth, it has a nontrivial optimization problem. In equilibrium, this family
effectively ignores matching considerations; i.e. s (0) 5 k ** (0). The reason is
reminiscent of why the least productive worker chooses no education in a
separating perfect Bayesian equilibrium in the Spence job market signalling
model. Clearly, s (0) $ k ** (0). Suppose the inequality is strict. By bequeathing
s (0), the family only ensures that the son matches with the woman with lowest
endowment. If instead the family bequeaths k ** (0), their utility from consumption
of the male good is increased, while the match is not adversely affected – with a
bequest of s (0) the son was the poorest male, and with a bequest of k ** (0) he is
still the poorest male. Thus, the poorest family will bequeath k ** (0). This yields
the initial value condition, s (0) 5 k ** (0). Since k 0 (0) . 0, the initial value
problem ((A.2) and s (0) 5 k ** (0)) has a unique increasing solution, s (i).29 Note
that s 9(i) → 1 ` as i → 0 (otherwise the poorest family would enjoy a first order
benefit in the quality of their son’s match by an infinitesimal increase in their
bequest, while only suffering a second order loss in the utility from consumption).
In summary, when k 0 (0) . 0, there is a unique equilibrium, and the equilibrium
bequest function solves the initial value problem described by (A.2) and s (0) 5
k ** (0).
Suppose now that k 0 (0) 5 0. Clearly, the bequest at zero must be zero; i.e.
s (0) 5 0. The initial value problem, however, described by (A.2) and the initial
28
Standard arguments show that any equilibrium s must be strictly increasing and differentiable on
(0,1).
29
The inverse initial value problem, where i(s ) is solved for as a function of the bequest s, is
Lipschitz in the relevant region. The technical issues here are similar to those that arise in the study of
separating equilibria in signalling games with a continuum of types – see Mailath (1987).
32
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
condition s (0) 5 0 is badly behaved: as i → 0, both u9(Ak 0 (i) 2 s (i)) and
Au9(As (i)) diverge to 1 `.
Rather than solving for equilibrium directly in this case, we obtain the
equilibrium as a limit by considering initial distributions of wealth k 0 1k,
] k] . 0,
and letting k] → 0. We have just argued that if the poorest family has strictly
positive wealth, then there is a unique equilibrium. Denote the equilibrium bequest
*
function for initial distributions of wealth k 0 1k] by s (i;k).
] Define s (i) 5
**
*
inf k s (i;k).
Since
s
(i;k)
[
[k
(i),Ak
(i)
1
Ak]
for
all
i,
s
is
well-defined.
0
]
]
]
]
*
Moreover, it satisfies (A.2) and so is differentiable.30 Since hs ( ? ;k)j
] and s are
continuous functions on the compact set [0,1] and the convergence of s ( ? ;k)
] to
s * is monotone (in fact, decreasing), the convergence is uniform.
We will take s * as the equilibrium bequest function in our two period example.
While we expect there are additional solutions to (A.2) satisfying s (0) 5 0, these
cannot be obtained as the limit of the unique equilibria of the game with
endowments k 0 (i) 1k] as ]k → 0. We call s * the robust bequest function (since it is
the only bequest function close to the equilibrium bequest functions for k] small).
As a final note on s * , note that, since s * (i) [ [k ** (i),Ai], ds * / di does not
diverge to 1 ` as i → 0.31
Now consider the general finite horizon model. In the last period, women
always accept the wealthiest suitor, just as in the last period of the two period
model. Bequest behavior in the penultimate period maximizes (A.1), where F is
now the distribution of wealth in the penultimate period. For each distribution of
wealth in this period, there is a unique equilibrium bequest function (if k 0 (0) 5 0,
there is a unique robust equilibrium bequest function). Since the utility of the
penultimate generation is thus completely determined by the bequest received (and
the distribution of wealth), women in the penultimate generation always accept the
wealthiest suitor. The argument can now be applied to earlier periods.
For the case k 0 (0) 5 0, the analysis of the previous paragraph relied on the
uniqueness of the robust bequest function in the two period model. If we do not
restrict attention to the robust bequest function, there are (potentially) multiple
equilibria in the two period game. This raises the possibility of constructing
additional equilibria of the three period model in which women do not always
30
Standard theorems on the continuity of solutions to differential equations cannot be applied, since
neither (A.2) nor its inverse are Lipschitz. However, for fixed i9 and e small (so that i9 2 e . 0), s 9(i;k)
]
converges uniformly to the right hand side of (A.2) for all i [ [i9 2 e,1]. [Note first that the family
hs ( ? ;k)j is equicontinuous, considered as functions with domain [i9 2 e,1]: there exists h . 0
]
(independent of k) such that if us 2 k ** (i;k)u , h, then s 9(i;k) . 2 dk ** (i;k) / di for all i [ [0,1], and so
]
]
]
]
there exists d . 0 (again, independent of k) such that s (i;k) $ k ** (i;k) 1 d for all i [ [i9 2 e,1] and all
]
]
]
k. Then pointwise convergence of s to s * is uniform on [i9 2 e,1], and so s * is uniformly
]
continuous.] This then implies that s * also satisfies (A.2) (see, for example, Rudin, 1976, Theorem
7.17).
31
In fact, for the case g 5 1 and k 0 (i) 5 i, using L’Hospital’s rule, it easy to show that lim ds * / di 5
Ab /(1 1 b ).
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
33
accept the wealthiest suitor, because the choice of suitor could affect which
equilibrium is played in the two period game. This type of construction underlies
the finite horizon folk theorem (see Benoit and Krishna, 1985, and Friedman,
1985). Note that the source of intertemporal incentives that supports women not
always accepting the wealthiest suitor relies on the bequest behavior of the entire
community changing in response to a deviation by one player.
As we saw, if k 0 (0) . 0, the equilibrium is necessarily unique and the slope of
the bequest function is infinite at zero. The nature of equilibria is thus qualitatively
different when k 0 (0) 5 0, in which case there may be multiple equilibria, and the
bequest functions in equilibria in which women always accept the wealthiest suitor
(we later refer to these as wealth-is-status equilibria) do not have infinite slope at
zero.
Finally, as the horizon becomes arbitrarily long, the equilibria converge to a
wealth-is-status equilibrium of the infinite horizon game. Fix the horizon length n.
Since in every equilibrium and in every period, women accept the wealthiest
suitor, the game can be viewed as a game where the set of players is the family
lines, the strategy space for family line i9 is the collection of sequences k ; hkt j nt 50
satisfying 0 # kt 11 # Akt and k0 5 k 0 (i9), and the payoff is
O b hu(Ak 2 k
n21
n
Ui 9 (k,hhk t (i)j t 50 : i ± i9j) ;
t
t
t 11
) 1 m t (kt )j
t 50
1 b n hu(Akn ) 1 m n (kn )j,
where m t (kt ) 5 lhi: k t (i) # kt j. The discussion above shows that this game has a
unique Nash equilibrium (unique robust equilibrium if k 0 (0) 5 0). Let
k tn : [0,1] → R 1 be the equilibrium bequests in period t , n of the n-period game.
`
For fixed t, the sequence hk tn j n5t
11 is nondecreasing and so Cauchy. Finally, since
this game is continuous at infinity (see Fudenberg and Levine, 1986), the limit of
the equilibria will be an equilibrium of the limit game, i.e. a wealth-is-status
equilibrium.
Appendix B
Proof of Proposition 3
The discussion before the statement of the proposition covers the necessity part.
We now prove the sufficiency part.
Recall that, because of (9), any offer acceptable to a female in the upper group
must involve a wealth level sufficiently higher than kˆ t (il ). Any such offer by
family line il involves a sacrifice by the parents (of utility order d ) that can only be
outweighed by the upper group female having an index bounded away from il .
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
34
Denote this bound by h . 0. Thus, it is enough to deter females with index
j [ (il 1 h,1], and any female in the upper group with an index j $ il 1 h suffers a
status loss of at least h per period.
Lemma 1. For all h . 0, there exists b 9 such that for b . b 9, no upper group
woman with index j [ (il 1 h,1] will deviate if the deviation results in a status of
no more than il .
Proof. A woman j in the upper group has an equilibrium payoff of (where k is the
wealth of her mate)
j
V * (k) 1 ]]].
(1 2 b )
An upper bound on her payoff from accepting increased consumption in this
period (via increased capital k . k) and losing the upper group status of her
descendants is
b il
b ( j 2 h)
V * (k ) 1 j 1 ]]] #V * (k ) 1 j 1 ]]].
(1 2 b )
(1 2 b )
This is not the best possible upper bound since in order to guarantee any status for
her descendants, the family line from this period will need to make wealth-isstatus consumption–savings decisions which yields a strictly lower discounted
stream of utility than V * (k ). Women in the upper group reject increased
consumption if
j
b ( j 2 h)
V * (k) 1 ]]] $V * (k ) 1 j 1 ]]],
(1 2 b )
(1 2 b )
which (since g . 1, V * (k ) , 0) is implied by
bh
V * (k) 1 ]]] $ 0,
(1 2 b )
or
c
O b H]]]
1 bhJ $ 0,
(1 2 g )
`
t
12g
t
(B.1)
t 50
where ct 5 l * A[(1 2 l * )A] t k.
¯
Fix e . 0. There exists c¯ such that, for all c $ c,
c 12g
]]] 1 bh . e.
(1 2 g )
¯ where
Since k 0 (il ) . 0, there exists T 1 such that ( l † A)[(1 2 l † )A] T 1 k 0 (il ) $ c,
l † 5 1 2 (A12g (2 / 3))1 / g (this is just l * with b 5 2 / 3). Then, if T $
H.L. Cole et al. / Journal of Public Economics 70 (1998) 5 – 35
35
maxhT 1 ,A1 / g (A 2 A1 / g )21 j, c t (il ) ; ( l * A)[(1 2 l * )A] t k 0 (il ) $ c¯ for all b $ 2 / 3
¯ then c t (i) $ c¯ for all
and t $ T. Moreover, since k 0 is nondecreasing, if c t (il ) $ c,
i . il .
Since wealth and consumption are increasing over time, if (B.1) is satisfied in
the initial period, then it will be satisfied in every period. If we define c †0 ;
l † Ak 0 (il ), then ct (i) $ c †0 . The left hand side of (B.1) can then be rewritten as
c
c
(c )
b
e
O b H]]]
1 bhJ 1 O b H]]] 1 bhJ $ T ]]] 1 ]]],
(1 2 g )
(1 2 g )
(1 2 g ) (1 2 b )
T
t
t50
12g
t
`
t
12g
t
† 12g
0
T 11
t 5T 11
which is positive for b sufficiently close to 1.
References
Benoit, J.-P., Krishna, V., 1985. Finitely repeated games. Econometrica 53, 905–922.
Brooks, N., 1995. Effects of community characteristics on community social behavior. PhD thesis,
University of Pennsylvania.
Burdett, K., Coles, M., 1997. Marriage and class. Quarterly Journal of Economics 112, 141–168.
Cole, H.L., Mailath, G.J., Postlewaite, A., 1992. Social norms, savings behavior, and growth. Journal of
Political Economy 100, 1092–1125.
Cole, H.L., Mailath, G.J., Postlewaite, A., 1995a. Incorporating concern for relative wealth into
economic models. Federal Reserve Bank of Minneapolis Quarterly Review 19, 12–21.
Cole, H.L., Mailath, G.J., Postlewaite, A., 1995b. Response to ‘Aristocratic equilibria’. Journal of
Political Economy 103, 439–443.
Friedman, J.W., 1985. Cooperative equilibria in finite horizon non-cooperative supergames. Journal of
Economic Theory 35, 390–398.
Fudenberg, D., Levine, D., 1986. Limit games and limit equilibria. Journal of Economic Theory 38,
261–279.
Landsburg, S.E., 1995. Aristocratic equilibria. Journal of Political Economy 103, 434–438.
Mailath, G.J., 1987. Incentive compatibility in signaling games with a continuum of types. Econometrica 55, 1349–1365.
Roth, A., Sotomayer, M.A.O., 1990. Two-Sided Matching. Cambridge University Press, Cambridge.
Rudin, W., 1976. Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York.
Stone, L., Stone, J.F., 1984. An Open Elite? England 1540–1880. Clarendon Press, Oxford.
Weiss, A., 1990. Efficiency Wages: Models of Unemployment, Layoffs, and Wage Dispersion.
Princeton University Press, Princeton.